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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228672, base_ring=CyclotomicField(1584))
M = H._module
chi = DirichletCharacter(H, M([0,891,0,416]))
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gp:[g,chi] = znchar(Mod(37, 228672))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228672.37");
| Modulus: | \(228672\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(25408\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1584\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | no, induced from \(\chi_{25408}(37,\cdot)\) |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{228672}(37,\cdot)\)
\(\chi_{228672}(541,\cdot)\)
\(\chi_{228672}(1693,\cdot)\)
\(\chi_{228672}(1837,\cdot)\)
\(\chi_{228672}(1909,\cdot)\)
\(\chi_{228672}(2701,\cdot)\)
\(\chi_{228672}(3205,\cdot)\)
\(\chi_{228672}(4645,\cdot)\)
\(\chi_{228672}(4861,\cdot)\)
\(\chi_{228672}(5293,\cdot)\)
\(\chi_{228672}(5509,\cdot)\)
\(\chi_{228672}(7165,\cdot)\)
\(\chi_{228672}(8677,\cdot)\)
\(\chi_{228672}(9253,\cdot)\)
\(\chi_{228672}(9397,\cdot)\)
\(\chi_{228672}(10117,\cdot)\)
\(\chi_{228672}(10333,\cdot)\)
\(\chi_{228672}(11125,\cdot)\)
\(\chi_{228672}(11197,\cdot)\)
\(\chi_{228672}(12205,\cdot)\)
\(\chi_{228672}(12997,\cdot)\)
\(\chi_{228672}(13573,\cdot)\)
\(\chi_{228672}(14365,\cdot)\)
\(\chi_{228672}(14653,\cdot)\)
\(\chi_{228672}(15661,\cdot)\)
\(\chi_{228672}(15877,\cdot)\)
\(\chi_{228672}(15949,\cdot)\)
\(\chi_{228672}(16237,\cdot)\)
\(\chi_{228672}(16453,\cdot)\)
\(\chi_{228672}(16741,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
| Field of values: |
$\Q(\zeta_{1584})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 1584 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((21439,185797,25409,169921)\) → \((1,e\left(\frac{9}{16}\right),1,e\left(\frac{26}{99}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 228672 }(37, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1307}{1584}\right)\) | \(e\left(\frac{23}{792}\right)\) | \(e\left(\frac{919}{1584}\right)\) | \(e\left(\frac{1525}{1584}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{877}{1584}\right)\) | \(e\left(\frac{773}{792}\right)\) | \(e\left(\frac{515}{792}\right)\) | \(e\left(\frac{889}{1584}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
